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105 relations: Abelian category, Abelian group, Additive category, Algebraic structure, Allegory (category theory), Antisymmetric relation, Associative property, Automorphism, Bilinear form, Binary operation, Binary relation, Bundle map, Cartesian closed category, Categories for the Working Mathematician, Category of abelian groups, Category of groups, Category of magmas, Category of manifolds, Category of metric spaces, Category of modules, Category of preordered sets, Category of relations, Category of rings, Category of sets, Category of small categories, Category of topological spaces, Category theory, Class (set theory), Closure (mathematics), Cokernel, Commutative diagram, Complete category, Complete partial order, Concrete category, Continuous function, Coproduct, Differentiable manifold, Directed graph, Discrete category, Endomorphism, Enriched category, Epimorphism, Equivalence of categories, Equivalence relation, Fiber bundle, Field (mathematics), Free category, Function (mathematics), Function composition, Functor, ..., Generator (mathematics), GNU Free Documentation License, Group (mathematics), Group action, Group homomorphism, Groupoid, Higher category theory, Identity (mathematics), Identity element, Identity function, Inverse element, Isomorphism, Kernel (category theory), Limit (category theory), Linear map, List of mathematical symbols, Loop (graph theory), Magma (algebra), Mathematics, Metric map, Metric space, Module (mathematics), Module homomorphism, Monoid, Monomorphism, Monotonic function, Morphism, Opposite category, Ordinal number, Partially ordered set, Preadditive category, Preorder, Product (category theory), Quantaloid, Reflexive relation, Ring (mathematics), Ring homomorphism, Samuel Eilenberg, Saunders Mac Lane, Scott continuity, Section (category theory), Semantics (computer science), Set (mathematics), Set theory, Stanford Encyclopedia of Philosophy, Subcategory, Topological space, Topos, Total order, Transitive relation, Tuple, Uniform continuity, Uniform space, Vector space, Vertex (graph theory). Expand index (55 more) »
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.
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Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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Additive category
In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts.
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Algebraic structure
In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.
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Allegory (category theory)
In the mathematical field of category theory, an allegory is a category that has some of the structure of the category of sets and binary relations between them.
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Antisymmetric relation
In mathematics, a binary relation R on a set X is anti-symmetric if there is no pair of distinct elements of X each of which is related by R to the other.
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Associative property
In mathematics, the associative property is a property of some binary operations.
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Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself.
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Bilinear form
In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map, where K is the field of scalars.
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Binary operation
In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set.
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Binary relation
In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.
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Bundle map
In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles.
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Cartesian closed category
In category theory, a category is considered Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors.
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Categories for the Working Mathematician
Categories for the Working Mathematician (CWM) is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg.
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Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms.
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Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms.
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Category of magmas
In mathematics, the category of magmas, denoted Mag, has as objects sets with a binary operation, and morphisms given by homomorphisms of operations (in the universal algebra sense).
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Category of manifolds
In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps.
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Category of metric spaces
In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms.
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Category of modules
In algebra, given a ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules.
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Category of preordered sets
The category Ord has preordered sets as objects and order-preserving functions as morphisms.
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Category of relations
In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms.
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Category of rings
In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (preserving the identity).
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Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets.
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Category of small categories
In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories.
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Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps or some other variant; for example, objects are often assumed to be compactly generated.
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Category theory
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
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Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.
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Closure (mathematics)
A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.
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Cokernel
In mathematics, the cokernel of a linear mapping of vector spaces f: X → Y is the quotient space Y/im(f) of the codomain of f by the image of f. The dimension of the cokernel is called the corank of f. Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel is a quotient object of the codomain (it maps from the codomain).
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Commutative diagram
The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result.
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Complete category
In mathematics, a complete category is a category in which all small limits exist.
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Complete partial order
In mathematics, the phrase complete partial order is variously used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties.
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Concrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, see Relative concreteness below).
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Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
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Coproduct
In category theory, the coproduct, or categorical sum, is a category-theoretic construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces.
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Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.
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Directed graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is a set of vertices connected by edges, where the edges have a direction associated with them.
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Discrete category
In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms: Since by axioms, there is always the identity morphism between the same object, we can express the above as condition on the cardinality of the hom-set Some authors prefer a weaker notion, where a discrete category merely needs to be equivalent to such a category.
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Endomorphism
In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself.
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Enriched category
In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category.
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Epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f: X → Y that is right-cancellative in the sense that, for all morphisms, Epimorphisms are categorical analogues of surjective functions (and in the category of sets the concept corresponds to the surjective functions), but it may not exactly coincide in all contexts; for example, the inclusion \mathbb\to\mathbb is a ring-epimorphism.
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Equivalence of categories
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same".
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Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
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Fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Free category
In mathematics, the free category or path category generated by a directed graph or quiver is the category that results from freely concatenating arrows together, whenever the target of one arrow is the source of the next.
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Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
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Function composition
In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.
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Functor
In mathematics, a functor is a map between categories.
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Generator (mathematics)
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts.
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GNU Free Documentation License
The GNU Free Documentation License (GNU FDL or simply GFDL) is a copyleft license for free documentation, designed by the Free Software Foundation (FSF) for the GNU Project.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Group homomorphism
In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".
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Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways.
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Higher category theory
In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.
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Identity (mathematics)
In mathematics an identity is an equality relation A.
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Identity element
In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.
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Identity function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.
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Inverse element
In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication.
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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Kernel (category theory)
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra.
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Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.
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Linear map
In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.
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List of mathematical symbols
This is a list of symbols used in all branches of mathematics to express a formula or to represent a constant.
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Loop (graph theory)
In graph theory, a loop (also called a self-loop or a "buckle") is an edge that connects a vertex to itself.
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Magma (algebra)
In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Metric map
In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous).
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Metric space
In mathematics, a metric space is a set for which distances between all members of the set are defined.
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Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
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Module homomorphism
In algebra, a module homomorphism is a function between modules that preserves module structures.
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Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.
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Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism.
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Monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.
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Morphism
In mathematics, a morphism is a structure-preserving map from one mathematical structure to another one of the same type.
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Opposite category
In category theory, a branch of mathematics, the opposite category or dual category Cop of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism.
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Ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.
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Partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.
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Preadditive category
In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups.
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Preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.
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Product (category theory)
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces.
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Quantaloid
In mathematics, a quantaloid is a category enriched over the category Sup of suplattices.
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Reflexive relation
In mathematics, a binary relation R over a set X is reflexive if every element of X is related to itself.
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Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
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Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.
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Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-born American mathematician who co-founded category theory with Saunders Mac Lane.
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Saunders Mac Lane
Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg.
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Scott continuity
In mathematics, given two partially ordered sets P and Q, a function f \colon P \rightarrow Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema, i.e. if for every directed subset D of P with supremum in P its image has a supremum in Q, and that supremum is the image of the supremum of D: that is, \sqcup f.
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Section (category theory)
In category theory, a branch of mathematics, a section is a right inverse of some morphism.
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Semantics (computer science)
In programming language theory, semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages.
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Set (mathematics)
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
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Set theory
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
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Stanford Encyclopedia of Philosophy
The Stanford Encyclopedia of Philosophy (SEP) combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users.
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Subcategory
In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms.
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Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
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Topos
In mathematics, a topos (plural topoi or, or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site).
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Total order
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.
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Transitive relation
In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.
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Tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements.
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Uniform continuity
In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between f(x) and f(y) cannot depend on x and y themselves.
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Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Vertex (graph theory)
In mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices).
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References
[1] https://en.wikipedia.org/wiki/Category_(mathematics)
